Restriction (mathematics)

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Restriction (mathematics)

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In mathematics, the restriction of a function refers to the process of limiting a function's domain to a proper subset while preserving its mapping behavior on that subset, resulting in a new function defined only on the chosen domain.^[1]^[2]^[3] Formally, given a function

f: X \to Y

and a subset

A \subseteq X

, the restriction of

f

A

, denoted

f|_A

f \upharpoonright A

, is the function

f|_A: A \to Y

such that

f|_A(x) = f(x)

for all

x \in A

.^[1]^[2]^[3] This construction ensures that the graph of the restricted function is the subset of the original graph consisting of pairs

(x, f(x))

where

x \in A

.^[2] Restrictions inherit several important properties from the original function. For instance, if

f

is injective (one-to-one) on

A

, then so is

f|_A

; similarly, continuity is preserved under restriction, meaning the restriction of a continuous function to a subset is continuous relative to the subspace topology.^[2] Restrictions are transitive: if

A \subseteq B \subseteq X

, then

(f|_B)|_A = f|_A

.^[2] Additionally, the preimage under the restriction satisfies

(f|_A)^{-1}(B) = A \cap f^{-1}(B)

for any

B \subseteq Y

.^[1] A classic example is the function

f: \mathbb{R} \to \mathbb{R}

defined by

f(x) = x^2

, which is not injective on all reals but becomes injective when restricted to the non-negative reals

[0, \infty)

, allowing an inverse given by the square root function.^[2] Another illustration is the absolute value function

f: \mathbb{Z} \to \mathbb{Z}

f(z) = |z|

, whose restriction to the natural numbers

\mathbb{N}

coincides with the identity function on

\mathbb{N}

.^[3] Beyond basic function theory, restrictions play key roles in advanced areas. In topology, the pasting lemma uses restrictions to continuous functions on subsets to establish continuity on their union under certain conditions.^[2] In algebra and database theory, they appear in selection operators that filter relations based on predicates.^[2] More abstractly, in sheaf theory, restrictions generalize to morphisms between sections over open sets in topological spaces, forming the foundation for coherent structures in algebraic geometry.^[2] The factorial function, for example, can be viewed as the restriction of the gamma function to positive integers.^[2]

Restriction of Functions

Formal Definition

In mathematics, the restriction of a function is a fundamental operation that limits the domain of the function to a specified subset while preserving its values on that subset. Formally, given sets

E

and

F

, and a function

f: E \to F

that is total on

E

(meaning it assigns a unique output in

F

to every element of

E

), the restriction of

f

to a subset

A \subseteq E

is the function

f|_A: A \to F

defined by

f|_A(x) = f(x)

for all

x \in A

.^[2]^[3]^[4] The notation for restriction commonly employs the vertical bar subscript, as in

f|_A

, though alternatives such as

f \restriction_A

f \upharpoonright A

are also used in some texts; regardless of the symbol, the codomain of the restricted function remains

F

unless explicitly modified.^[2]^[3] This construction assumes that

E

and

F

are arbitrary sets and that

f

is well-defined on the entire domain

E

, ensuring the restriction is itself a total function on

A

.^[4]

Examples

A classic example of function restriction is the squaring function

f(x) = x^2

, originally defined on the domain

\mathbb{R}

with codomain

\mathbb{R}

. This function is neither injective nor surjective over

\mathbb{R}

, as

f(-1) = f(1) = 1

and negative values are missed in the image. Restricting the domain to the non-negative reals

[0, \infty)

yields

f|_{[0, \infty)}: [0, \infty) \to [0, \infty)

, which is now bijective, as it is strictly increasing and onto its image.^[5] In analytic contexts, the factorial function on positive integers

\mathbb{N}

arises as a restriction of the gamma function

\Gamma(z)

, defined for complex

z

with positive real part via the integral

\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt

. Specifically,

\Gamma(n+1) = n!

for

n \in \mathbb{N}

, providing a continuous interpolation of the discrete factorial values.^[6] For trigonometric functions, consider the sine function

\sin x

\mathbb{R}

, which is periodic and thus not injective. Restricting the domain to

[-\pi/2, \pi/2]

produces

\sin|_{[-\pi/2, \pi/2]}: [-\pi/2, \pi/2] \to [-1, 1]

, a bijective mapping essential for defining the principal branch of the inverse sine.^[7] In general, selecting appropriate domain subsets via restriction can transform a function's properties, such as rendering it monotonic where it was not originally; for instance, the squaring function gains strict monotonicity on

[0, \infty)

despite decreasing on

(-\infty, 0]

.^[5]

Properties

The restriction operation satisfies certain identity and transitivity properties. Specifically, for a function

f: E \to F

, the restriction to the entire domain

E

yields the original function, i.e.,

f|_E = f

.^[5] Furthermore, successive restrictions are equivalent to a single restriction: if

B \subseteq A \subseteq E

, then

(f|_A)|_B = f|_B

.^[2] Restriction preserves several key algebraic and analytic properties of functions. If

f: E \to F

is injective, then its restriction

f|_A: A \to F

to any subset

A \subseteq E

is also injective, since distinct elements in

A

map to distinct elements in

F

by the original injectivity.^[8] Similarly, if

f

is continuous (in the topological sense), the restriction to a subspace

A

is continuous relative to the subspace topology on

A

.^[9] For real-valued functions, if

f

is differentiable on an open set

E \subseteq \mathbb{R}

, then

f|_A

is differentiable at interior points of

A \subseteq E

, with the derivative matching that of

f

.^[10] Monotonicity is also preserved: if

f

is monotone on

E

, its restriction to

A \subseteq E

remains monotone on

A

.^[11] However, surjectivity is not generally preserved under restriction. While

f|_A

is surjective onto its image

f(A) \subseteq F

, it may fail to be surjective onto the full codomain

F

unless

f(A) = F

.^[5] Regarding composition, the restriction of a composite function aligns with the composition of restrictions under compatible domains. For functions

f: E \to G

and

g: G \to F

with

A \subseteq E

, the restriction satisfies

(g \circ f)|_A = g|_{f(A)} \circ (f|_A)

.^[2]

Extensions of Functions

Definition

In mathematics, particularly in set theory and topology, an extension of a function

f: A \to F

is a function

\tilde{f}: E \to F

defined on a superset

E \supseteq A

such that the restriction of

\tilde{f}

A

coincides with

f

, i.e.,

\tilde{f}|_A = f

.^[12] This operation serves as the inverse process to restricting a function's domain, enlarging it while preserving the original mapping on the initial domain.^[13] Such extensions are not unique in general, since the values of

\tilde{f}

E \setminus A

may be defined arbitrarily within the codomain

F

, subject only to the consistency requirement on

A

.^[12] However, uniqueness or existence can hold under additional conditions; for instance, in topology, the Tietze extension theorem guarantees the existence of a continuous extension

\tilde{f}: X \to \mathbb{R}

for a continuous

f: A \to \mathbb{R}

, where

A

is a closed subspace of a normal topological space

X

.^[13] This theorem extends to bounded real-valued functions with range preserved in a closed interval

[a, b]

.^[12] Partial extensions refer to cases where the enlargement of the domain maintains specific properties of the original function, such as continuity or differentiability.^[13] For continuous functions on closed subsets of normal spaces, the Tietze theorem ensures such a continuous partial extension exists to the entire space.^[12]

Relation to Restrictions

In the theory of function extensions, compatibility with restrictions is fundamental. Every restriction of a function to a subset of its domain admits trivial extensions to a larger domain; for instance, one can define the function to take a constant value on the complement of the subset while agreeing with the original on the subset.^[14] Such extensions always exist for arbitrary functions, as the values on the complement can be chosen freely without additional constraints. However, non-trivial extensions that preserve properties like continuity depend on the extendability of the restricted function. For continuous functions on closed subsets of normal topological spaces, the Tietze extension theorem ensures the existence of a continuous extension to the entire space.^[15] The relation between extensions and restrictions also appears in iterative processes, where successive restrictions to subsets followed by extensions can recover the original function under suitable conditions. For example, for any function

f: [0,1]^2 \to \mathbb{R}

, there exists a c-dense set

D \subset [0,1]^2

such that the restriction

f|_D

is separately continuous, and this restriction admits an extension

h

[0,1] \times [0,1]

that is quasi-continuous and agrees with

f

on a set containing

D

, thereby approximating or recovering key behaviors of the original

f

.^[16] This iterative interplay highlights how restrictions can be used to simplify analysis while extensions allow reconstruction, provided the function satisfies properties like separate continuity on dense subsets. A illustrative example of the challenges in extending restrictions while preserving functional properties is the square root function

\sqrt{x}

defined on $[0, \infty)$ . There is no real-valued continuous extension to $\mathbb{R}$ that satisfies $f(x)^2 = x$ for all $x \in \mathbb{R}$ , since for $x < 0$ , no real number squares to a negative value.^[17] However, piecewise definitions enable extensions that agree with $\sqrt{x}$

on $[0, \infty)$ but define different behaviors on $(-\infty, 0)$ , such as setting $f(x) = 0$ for $x < 0$ , yielding a continuous (though non-injective) function on $\mathbb{R}$ . Historically, the ideas of restricting and extending functions emerged in 18th-century analysis, particularly in addressing partial differential equations, where mathematicians like Euler broadened the notion of functions to include piecewise twice-differentiable forms to solve equations such as the wave equation $\partial^2 y / \partial t^2 = c^2 \partial^2 y / \partial x^2$ .^[18] This allowed extensions of solutions from initial or boundary data to larger domains, facilitating developments in heat conduction and vibration problems by Fourier and others in the 19th century.

Applications of Function Restrictions

Inverse Functions

In mathematics, restricting the domain of a non-injective function can yield an injective restriction, facilitating the definition of an inverse function. Consider a function

f: E \to F

that fails to be injective over its full domain

E

. A subset

A \subseteq E

can be selected such that the restriction

f|_A: A \to F

is injective. To construct the inverse, the codomain is adjusted to the image

f(A) \subseteq F

, rendering

f|_A: A \to f(A)

bijective. The inverse then exists as

(f|_A)^{-1}: f(A) \to A

, satisfying

(f|_A)^{-1}(f(a)) = a

for all

a \in A

. This approach is essential for defining inverses in contexts where the original function maps multiple domain elements to the same codomain element.^[19] A representative example is the squaring function

f(x) = x^2: \mathbb{R} \to \mathbb{R}

, which is not injective because

f(-x) = f(x)

for

x \neq 0

. Restricting the domain to the non-negative reals gives

f|_{[0, \infty)}: [0, \infty) \to [0, \infty)

, which is bijective since it is strictly increasing on this interval. The inverse is the principal square root function, defined by

(f|_{[0, \infty)})^{-1}(y) = \sqrt{y}, \quad y \in [0, \infty),

(f∣[0,∞))−1(y)=y

,y∈[0,∞), satisfying $\sqrt{x^2} = x$

=x for $x \geq 0$ and $(\sqrt{y})^2 = y$

)2=y for $y \geq 0$ . This restriction preserves the function's essential behavior while enabling invertibility.^[19] For the inverse to be bijective, the restricted codomain must be precisely the image $f(A)$ , ensuring surjectivity: every element in $f(A)$ is hit exactly once from $A$ . The domain of the inverse is $f(A)$ , with range $A$ , guaranteeing the identity mappings hold: $f|_A \circ (f|_A)^{-1} = \mathrm{id}_{f(A)}, \quad (f|_A)^{-1} \circ f|_A = \mathrm{id}_A.$ Without this codomain adjustment, the restriction might be injective but not surjective onto $F$ , precluding a full inverse. This setup aligns the domain and range appropriately for composition properties.^[19] In general settings, maximal injective restrictions address the challenge of selecting optimal subsets $A$ , where $A$ is inclusion-maximal among all subsets of $E$ on which $f$ is injective—no proper superset of $A$ admits an injective restriction of $f$ . Such maximal subsets exist under the axiom of choice: consider the partially ordered set of all pairs $(B, f|_B)$ where $B \subseteq E$ and $f|_B$ is injective, ordered by extension (i.e., $(B, f|_B) \leq (C, f|_C)$ if $B \subseteq C$ and $f|_C|_B = f|_B$ ). Every chain in this poset has an upper bound (the union), so Zorn's lemma yields a maximal element, providing a maximal domain for injectivity and thus a corresponding inverse on the image. These maximal restrictions maximize the scope of the inverse but may lack the symmetry or simplicity of ad hoc choices like principal branches in calculus.^[20]

Pasting Lemma

The pasting lemma, a key result in general topology concerning the restriction and extension of continuous functions, asserts that continuous functions defined on the pieces of an open cover of a topological space can be glued together to form a continuous function on the entire space, provided they agree on the overlaps. Specifically, let

X

be a topological space and

\{U_i\}_{i \in I}

an open cover of

X

, where

I

is an index set. Suppose

Y

is another topological space, and for each

i \in I

, there is a continuous function

f_i: U_i \to Y

. If

f_i

and

f_j

agree on

U_i \cap U_j

for all

i, j \in I

, then there exists a unique continuous function

f: X \to Y

such that

f|_{U_i} = f_i

for each

i \in I

. This function

f

is defined by

f(x) = f_i(x)

for any

x \in U_i

. A proof sketch relies on the definition of continuity in terms of preimages of open sets. To show

f

is continuous, let

V \subseteq Y

be open. Then

f^{-1}(V) = \bigcup_{i \in I} f_i^{-1}(V)

. For each

i

f_i^{-1}(V)

is open in

U_i

by continuity of

f_i

, and since

U_i

is open in

X

f_i^{-1}(V)

is open in

X

. The union of open sets is open, so

f^{-1}(V)

is open in

X

. The pasting lemma appears in mid-20th century topology texts as a standard tool for constructing continuous maps, with early presentations in works like John L. Kelley's General Topology (1955). A variant addresses pasting over closed sets: if

\{C_i\}_{i \in I}

is a finite collection of closed subsets covering

X

, and continuous functions

g_i: C_i \to Z

(for topological space

Z

) agree on pairwise intersections, then the glued function

g: X \to Z

is continuous. For infinite closed covers, additional conditions such as the space being normal are required to ensure the result holds in general.

Sheaves

In sheaf theory, a branch of mathematics central to algebraic geometry and topology, restriction maps form the backbone of the structure that coordinates local information across open subsets of a topological space into a cohesive global object. Introduced by Jean Leray during his internment as a prisoner of war in the early 1940s and formalized in subsequent works, a sheaf

\mathcal{F}

of sets (or more generally, of abelian groups or modules) on a topological space

X

assigns to each open subset

U \subseteq X

a set

\mathcal{F}(U)

, called the sections over

U

, together with restriction maps

\rho_{V,U}: \mathcal{F}(U) \to \mathcal{F}(V)

for every open

V \subseteq U

. These maps satisfy the sheaf axioms: locality, which states that two sections over

U

that restrict to the same section on every member of an open cover of

U

must be equal, and gluing, which asserts that compatible sections over the members of an open cover (agreeing on pairwise intersections) can be uniquely glued to a global section over the union.^[21]^[22] Unlike simple domain restrictions of functions, the restriction maps in sheaves function as morphisms in a presheaf category, preserving the algebraic or topological structure of the sections while enforcing consistency across the space. This setup ensures that the sheaf captures "local-to-global" behavior, where data defined patch-wise must cohere under restrictions, making sheaves ideal for studying phenomena like cohomology groups that vanish locally but not globally. The gluing axiom extends classical results such as the pasting lemma for continuous functions, providing an axiomatic framework for such compatibilities in more abstract settings.^[22]^[23] A concrete illustration is the constant sheaf

\underline{S}

associated to a discrete set

S

X

, where sections over an open

U

consist of locally constant functions from

U

S

; the restriction maps act pointwise, projecting the function values while maintaining local constancy. For instance, on a connected open set, sections are genuine constant functions to

S

, but on disconnected components, they allow varying constants that agree under restrictions to smaller opens. This example highlights how restrictions enforce the sheaf's coherence, ensuring that local variations remain trivial in the constant case.^[23] Since the 1960s, restriction maps within sheaf theory have underpinned modern developments like étale cohomology, pioneered by Alexander Grothendieck to equip algebraic varieties with a Weil cohomology theory via the étale topology, where restrictions respect finite étale covers rather than just opens.^[24]

Restriction of Relations

Left and Right Restrictions

In the context of binary relations, the left restriction (also known as domain restriction) of a relation

R \subseteq A \times B

to a subset

A' \subseteq A

is defined as the relation

R \lrestriction_{A'} = \{ (a, b) \in A' \times B \mid (a, b) \in R \}

.^[2]^[25] This operation retains only those pairs in

R

where the first component lies in

A'

, effectively narrowing the domain while preserving the full codomain

B

. Similarly, the right restriction (or range restriction) to a subset

B' \subseteq B

{}_{B'} \lrestriction R = \{ (a, b) \in A \times B' \mid (a, b) \in R \}

, which limits the codomain to

B'

while keeping the original domain

A

.^[2]^[25] These definitions parallel the domain and codomain restrictions for functions but apply to arbitrary relations, allowing pairs where the second component may not be uniquely determined by the first.^[2] The notation

\lrestriction

is commonly employed for left restrictions, with the subscript indicating the restricting subset, while right restrictions may use a prefixed subscript or analogous symbols such as

\triangleright

in some texts.^[2] For example, consider the relation

R = \{ (1, a), (2, a), (2, b), (3, c) \} \subseteq \{1,2,3\} \times \{a,b,c\}

; the left restriction to

A' = \{2,3\}

yields

R \lrestriction_{A'} = \{ (2, a), (2, b), (3, c) \}

, and the right restriction to

B' = \{a, c\}

gives

{}_{\{a,c\}} \lrestriction R = \{ (1, a), (2, a), (3, c) \}

.^[25] For n-ary relations, restrictions extend naturally by applying the binary case to individual coordinates, often via projections to that coordinate. For an n-ary relation

R \subseteq A_1 \times \cdots \times A_n

, the restriction to a subset

S_k \subseteq A_k

in the k-th position is

\{ (a_1, \dots, a_n) \in A_1 \times \cdots \times S_k \times \cdots \times A_n \mid (a_1, \dots, a_n) \in R \}

, which can be computed iteratively by projecting onto the relevant binary slices if needed.^[2] This approach allows sequential restrictions across multiple coordinates, maintaining the structure of the original relation. Left and right restrictions preserve key properties of the original relation, particularly for partial orders. If

R

is reflexive, then

R \lrestriction_{A'}

is reflexive on

A'

; likewise, transitivity is preserved, as chains within the restricted domain remain valid under the original relation. For a partial order on a set, the left restriction to a subset induces a partial order on that subset, retaining reflexivity, antisymmetry, and transitivity. The same holds for right restrictions when the relation is homogeneous (domain equals codomain).

Anti-Restrictions

In the context of binary relations, anti-restrictions provide a means to subtract specified subsets from the domain or codomain, effectively excluding certain pairs while preserving the remainder of the relation. For a binary relation

R \subseteq A \times B

, the domain anti-restriction to a subset

A' \subseteq A

is defined as

R \setminus (A' \times B)

, which removes all pairs whose first component lies in

A'

. This operation yields a new relation

R' \subseteq (A \setminus A') \times B

whose domain is contained in the complement of

A'

. Similarly, the range anti-restriction to

B' \subseteq B

R \setminus (A \times B')

, resulting in a relation

R'' \subseteq A \times (B \setminus B')

that excludes pairs with second components in

B'

. These definitions appear in formal specification frameworks, where they facilitate precise manipulation of relational structures by exclusion rather than inclusion.^[26] Notation for anti-restrictions varies but commonly employs set difference or specialized symbols to distinguish from standard restrictions. The explicit set subtraction

\setminus

highlights the removal process, as in the definitions above. In Z notation, a standard formal language for specifying mathematical structures, domain anti-restriction is denoted

A' \triangleright R

, equivalent to

R \setminus (A' \times \ran(R))

, and range anti-restriction as

R \triangleright B'

, or

R \setminus (\dom(R) \times B')

, where

\dom

and

\ran

denote the domain and range of the relation. These notations emphasize the antithetical nature to restriction operators like

A' \triangleleft R

(domain restriction) and

R \triangleleft B'

(range restriction), which intersect with subsets rather than subtract. For instance, in an equivalence relation on a set of data records, an anti-restriction might remove pairs involving deprecated records (e.g.,

A'

as invalid keys), yielding a reduced relation that excludes obsolete equivalences without altering surviving pairs.^[26] Unlike standard restrictions, which generally preserve relational properties such as reflexivity, symmetry, and transitivity when the restricting set aligns with the relation's structure, anti-restrictions can disrupt these properties by asymmetrically removing pairs. Consider a symmetric relation

R

on a set

X

, where

(x, y) \in R

implies

(y, x) \in R

. Applying a domain anti-restriction with

A' = \{a\} \subseteq X

removes all pairs

(a, y) \in R

, but if some

(y, a)

with

y \notin A'

remains, the resulting relation loses symmetry since the converse pair is absent. This disruption arises because anti-restriction targets one side of the Cartesian product without guaranteeing balanced removal, contrasting with the preservative behavior of intersection-based restrictions. In the case of transitivity, removing domain elements may sever chains of related pairs, breaking the property even if the original relation was transitive. Such effects are noted in analyses of relational operations, where anti-restrictions are used judiciously to avoid unintended structural changes.^[26] Anti-restrictions extend naturally to multi-relations, where elements in the domain may relate to multiple codomain elements, by applying the subtraction uniformly to all affected pairs. For example, given a multi-relation

R

representing possible transitions in a state machine (with multiple outputs per input), a domain anti-restriction to exclude faulty states

A'

removes all outgoing transitions from those states, preserving the multi-valued nature for remaining states while simplifying the model. This operation connects to quotient constructions in relation theory, where anti-restriction can eliminate pairs involving representatives of equivalence classes to induce a well-defined relation on the quotient set; for instance, in modular equivalence, anti-restricting by residue classes modulo

n

aids in projecting the relation onto the quotient space

\mathbb{Z}/n\mathbb{Z}

, ensuring compatibility with the partition. These applications underscore anti-restrictions' role in refining relations for modular or reduced structures without introducing extraneous pairs.^[26]

Selection Operators

In relational algebra, the selection operator

\sigma_P(R)

restricts a relation

R

to the subset of tuples that satisfy a specified predicate

P

, effectively filtering rows based on logical conditions.^[27] This operation parallels conditional domain restriction by selecting only those elements meeting the criterion, preserving the relation's schema while reducing its cardinality.^[27] The predicate

P

can involve comparisons, logical connectives, or arithmetic on attributes, enabling precise data subsetting.^[27] The notation for the selection operator typically places the predicate as a subscript, such as

\sigma_{\text{age > 30}}(\text{Employees})

, which yields all tuples from the Employees relation where the age attribute exceeds 30.^[27] Formally, its semantics are defined as

\sigma_P(R) = \{ t \in R \mid P(t) = \true \}

, ensuring the output remains a valid relation with the same attributes as the input.^[27] This syntax originated in the evolution of relational algebra following Edgar F. Codd's 1970 introduction of restriction operations in his relational model, where selections were initially generalized using auxiliary relations for pattern matching.^[28] Key properties of the selection operator include its commutativity with the projection operator

\pi

, provided the predicate

P

depends solely on the projected attributes:

\pi_a(\sigma_P(R)) = \sigma_P(\pi_a(R))

for attribute set

a

.^[27] It also exhibits closure under composition, as

\sigma_P(\sigma_Q(R)) = \sigma_{P \land Q}(R)

, allowing multiple filters to combine into a single, equivalent selection for query optimization.^[27] These algebraic laws facilitate efficient query rewriting in database systems.^[27] In modern implementations, the selection operator corresponds directly to the WHERE clause in SQL, which applies predicates to filter rows during query execution.^[29] For instance, the SQL query SELECT * FROM Employees WHERE age > 30 translates to

\sigma_{\text{age > 30}}(\text{Employees})

in relational algebra.^[29] This integration, rooted in Codd's foundational work, underpins the declarative nature of SQL queries in relational database management systems.^[28]

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